googologywikiaorg-20200223-history
User blog:Edwin Shade/On The Formalization, Oxymoronical Over-complicated Elucidification and Parameterization Of A Transfinite Generalization Of Taranovsky’s Notation, (O.T.F.O.O.E.A.P.O.F.T.G.O.T. for short)
In this blog post I will posit, elucidate, and analyze an intuitively graspable generalization of Taranovsky’s ordinal notation which provably supersedes Taranovsky’s original notation by transfinite diagonalization over the ordering of entries in expressions of the form \(C(\alpha_1,\alpha_2)\), whilst preserving the notions of ‘’standard form’’ and multiple cardinalities. To begin with it will be of great assistance if in our mental cogitation we are able to formulate a visualization of the notations about which we speak, so that by juxtaposition of the two formulations, one visual, the other formal, we thereby come to a fuller understanding of the complete work. Note that in Taranovsky’s notation, (in which familiarity is hereby assumed), any conceivable expression will always be of the form \(C(\alpha_1,\alpha_2)\), in which \(\alpha_1\) and \(\alpha_2\) are either base constituents of Taranovsky’s notation escaping capsulation through lesser expressions, (i.e. \(C(0,0)\), \(\Omega_n\) where \(n\in\mathbb{N^+}\) ), or are merely other expressions of the form \(C(\alpha_1,\alpha_2)\) themselves. Thus by considering the expression \(C(\alpha_1,\alpha_2)\) as isomorphic to a vertically bipartioned equiangular quadrilateral containing two smaller visual constructions isomorphic to \(\alpha_1\) and \(\alpha_2\), from left to right respectively, we may progressively build a visual representation of any conceivable expression within Taranovsky’s notation. \(C(0,0)\) can be modeled with an empty vertically bipartioned equiangular quadrilateral, and \(\Omega_n\) can be modeled with multiple colors for each cardinality, black for \(\Omega_1\), crimson for \(\Omega_2\), cobalt for \(\Omega_3\), yellow for \(\Omega_4\), green for \(\Omega_5\), and so on as you desire. The end pictorial product pleasantly produces a Mondrian-esque pallette. Here are visualizations of Taranovsky expressions in union with the corresponding string and ordinal, (when possible). Now I would like to introduce a less powerful version of Taranovsky's notation, in which every valid expression is of the form \(C(\alpha)\), and is a less powerful edition of the original Taranovsky notation. The terms in this notation I have called "Tier-1 Tarrack's", standing for "Tier-1 Taranovsky Racks", where the tier denoted the number of entries in the rack, or expression. All content below this line will be expanded and clarified in time, with pictures to help those who wish for a more visual explanation. ---- As is expected, the limit of Tier-1 Tarracks is ω2, where as the limit of Tier-2 Tarracks, (traditional T.O.N.), is as of yet unknown. Tier-3 Tarracks function similarly to Tarracks of a lesser Tier, but there are three entries and the limit of the expression \(C(0,C(0,C(0,C(0,...C(0,0,0)...,0),0),0),0)\) is \(C(0,\Omega,0\). Logically, as you can have tier-3 Tarracks you may have tier-4 Tarracks and so forth, even going as far as to have Tarracks with a transfinite level of tiers. Let \(\mathbb{T}^{\mathfrak{L}}_{\alpha}\) denote the limit of tier-\(\alpha\) Tarracks. a work in progress, which means I can't finish this in one day, but will nonetheless be contributing to this blog post frequently. I haven't yet reached the transfinite generalization yet, but when I do it will be agreed upon it is very powerful. ---- My Replies To Comments in The Comments Section I will posting my replies to the comments I received to this blog post here, as for whatever reason attempting to respond now makes this pop up on my screen: To Boboris02: First Comment: Thanks, I intend to continue my first blog post on Taranovsky's notation, but I will be typing it on the wiki part-by-part as I further the analysis on paper. To Cloudy176: First Comment: What you did is pretty cool. I'm not experienced with programming, but I roughly understand what the commands in the source code stand for, (e.g. is for text, creates a table, and is most likely a division line, etc...). I'd like to expand on what you've done and see if I can create a larger table this time. Second Comment: Yes, I think that's true too. The title has disadvantages then, though I choose it to satire the long names of research essays I've found in magazines I have. Category:Blog posts